We study the synchronization of Kuramoto oscillators on networks where only a fraction of them is subjected to a periodic external force. When all oscillators receive the external drive the system always synchronize with the periodic force if its intensity is sufficiently large. Our goal is to understand the conditions for global synchronization as a function of the fraction of nodes being forced and how these conditions depend on network topology, strength of internal couplings and intensity of external forcing. Numerical simulations show that the force required to synchronize the network with the external drive increases as the inverse of the fraction of forced nodes. However, for a given coupling strength, synchronization does not occur below a critical fraction, no matter how large is the force. Network topology and properties of the forced nodes also affect the critical force for synchronization. We develop analytical calculations for the critical force for synchronization as a function of the fraction of forced oscillators and for the critical fraction as a function of coupling strength. We also describe the transition from synchronization with the external drive to spontaneous synchronization.
Synchronization plays a key role in information processing in neuronal networks. Response of specific groups of neurons are triggered by external stimuli, such as visual, tactile or olfactory inputs. Neurons, however, can be divided into several categories, such as by physical location, functional role or topological clustering properties. Here we study the response of the electric junction C. elegans network to external stimuli using the partially forced Kuramoto model and applying the force to specific groups of neurons. Stimuli were applied to topological modules, obtained by the ModuLand procedure, to a ganglion, specified by its anatomical localization, and to the functional group composed of all sensory neurons. We found that topological modules do not contain purely anatomical groups or functional classes, corroborating previous results, and that stimulating different classes of neurons lead to very different responses, measured in terms of synchronization and phase velocity correlations. In all cases, however, the modular structure hindered full synchronization, protecting the system from seizures. More importantly, the responses to stimuli applied to topological and functional modules showed pronounced patterns of correlation or anti-correlation with other modules that were not observed when the stimulus was applied to ganglia.Understanding the network of neuronal connections in the brain is key to unravel the way it works and processes information. The complexity of these networks has been emphasized by many authors [1], and characterized with different measures, such as degree distribution, transitivity and betweenness centrality [2]. An important feature of neural networks is their high degree of heterogeneity, in the sense that the number of connections per neuron varies considerably and typically displays some sort of power law distribution. Moreover, neurons tend to form communities, where the density of connections is higher within than among communities. Because connections are constrained by anatomical features, neurons are also organized into physically arranged clusters, such as lobes or ganglia, where neurons with different functional roles coexist [3][4][5].Communities are often related to specialized areas of the brain and their number and structure are an indication of how many different tasks it can perform [6]. The integration of communities, on the other hand, measures how well the outcomes of these different processes can combined to build a global view of the inputs [3]. When triggered by external stimuli, such as visual or olfactory inputs, the information processing occurs by the synchronized firing of neurons responsible to process those specific tasks [7,8]. Synchronization of larger sets of neurons, or even global synchronization, indicates cerebral disorders [9] such as epilepsy [10] and Alzheimer's disease [11], causing a general breakdown in the neuronal network. Lack of synchronization, on the other hand, suggests difficulty to respond to the stimulus or to function properly,...
We study a binary dynamical process that is a representation of the voter model with two candidates and opinion makers. The voters are represented by nodes of a network of social contacts with internal states labeled 0 or 1 and nodes that are connected can influence each other. The network is also perturbed by opinion makers, a set of external nodes whose states are frozen in 0 or 1 and that can influence all nodes of the network. The quantity of interest is the probability of finding m nodes in state 1 at time t. Here we study this process on star networks, which are simple representations of hubs found in complex systems, and compare the results with those obtained for networks that are fully connected. In both cases a transition from disordered to ordered equilibrium states is observed as the number of external nodes becomes small. For fully connected networks the probability distribution becomes uniform at the critical point. For star networks, on the other hand, we show that the equilibrium distribution splits in two peaks, reflecting the two possible states of the central node. We obtain approximate analytical solutions for the equilibrium distribution that clarify the role of the central node in the process. We show that the network topology also affects the time scale of oscillations in single realizations of the dynamics, which are much faster for the star network. Finally, extending the analysis to two stars we compare our results with simulations in simple scale-free networks.
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